In this paper, we develop a geometry-inspired methodology for generating systematic and structured Grassmannian constellations with large cardinalities. In the proposed methodology we begin with a small close-to-optimal 'parent' Grassmann constellation. Each point in this constellation is augmented with a number of 'children' points, which are generated along a set of geodesics emanating from that point. These geodesics are chosen to ensure close-to-maximal spacing. In particular, the directions of the geodesics and the distance that each 'children' point is moved are chosen to maximize the pairwise Frobenius distance between the resulting constellation points. Although finding these directions directly seems difficult, by embedding the Grassmann manifold on a sphere of larger dimension, we were able to develop structures that are not only simple to generate but that also yield constellations that, under certain conditions, satisfy the maximum distance criterion and lie within a decaying gap from a tight upper bound. Numerical results suggest that the performance of the new constellations is comparable to that of the ones generated directly and significantly better than the performance of the ones generated using the exponential map.

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Conference 2016 IEEE International Symposium on Information Theory, ISIT 2016
Attiah, K.M. (Kareem M.), Seddik, K. (Karim), Gohary, R, & Yanikömeroǧlu, H. (2016). A systematic design approach for non-coherent Grassmannian constellations. In IEEE International Symposium on Information Theory - Proceedings (pp. 2948–2952). doi:10.1109/ISIT.2016.7541839